Effects of eddy currents in transformer windings
01 Aug 1966-Vol. 113, Iss: 8, pp 1387-1394
TL;DR: In this article, the effect of eddy currents on transformer windings is considered and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings.
Abstract: The effects of eddy currents in transformer windings are considered, and a method is derived for calculating the variation of winding resistance and leakage inductance with frequency for transformers with single-layer, multilayer and sectionalised windings. The method consists in dividing the winding into portions, calculating the d.c. resistances and d.c. leakage inductances of each of these portions, and then multiplying the d.c. values by appropriate factors to obtain the corresponding a.c. values. These a.c. values are then referred to, say, the primary winding and summed to give the total winding resistance and leakage inductance of the transformer. Formulas are derived and quoted for calculating the d.c. resistances and leakage inductances of the winding portions. Theoretical expressions are derived for the variation with frequency etc. of the factors by which the d.c. values must be multiplied to obtain the corresponding a.c. values. These expressions are presented in the form of graphs, permitting the factors to be read as required.
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TL;DR: In this paper , the authors derived a simple analytical model to calculate the total losses of a multi-stranded cable carrying a Direct Current (DC) affected by a high-frequency ripple.
Abstract: In electric vehicles, currents with high-frequency ripples flow in the power cabling system due to the switching operation of power converters. Inside the cables, a strong coupling between the thermal and electromagnetic phenomena exists, since the temperature and Alternating Current (AC) density distributions in the strands affect each other. Due to the different time scales of magnetic and heat flow problems, the computational cost of Finite Element Method (FEM) numeric solvers can be excessive. This paper derives a simple analytical model to calculate the total losses of a multi-stranded cable carrying a Direct Current (DC) affected by a high-frequency ripple. The expression of the equivalent AC cable resistance at a generic frequency and temperature is derived from the general treatment of multi-stranded multi-layer windings. When employed to predict the temperature evolution in the cable, the analytical model prevents the use of complex FEM models in which multiple heat flow and magnetic simulations have to be run iteratively. The results obtained for the heating curve of a 35 mm2 stranded cable show that the derived model matches the output of the coupled FEM simulation with an error below 1%, whereas the simple DC loss model of the cable gives an error of 2.4%. While yielding high accuracy, the proposed model significantly reduces the computational burden of the thermal simulation by a factor of four with respect to the complete FEM routine.
2 citations
Dissertation•
01 Jan 2017
TL;DR: A new methodology, based on the multiple scales method (MSM), is introduced which homogenises the complex windings domain and allows for the estimation of its effective thermal properties and enables improved information on hotspot location and magnitude.
Abstract: Temperature is one of the parameters that limits the output torque and reduces the lifespan of electrical machines. Models that can provide accurate estimation of the temperature field in the most critical components (e.g. windings) at lower computational effort can be useful to improve the design process and reduce the time to market. Depending on the application, engineers usually rely on hi-fidelity models, e.g. based on the finite elements method (FEM), or lower order models, e.g. thermal equivalent circuits (TECs). The aim of the present work is to provide new tools and methodologies to obtain the temperature distribution within the windings using reduced order hi-fidelity models or improved TEC that could account for any working condition, including AC effects. A new methodology, based on the multiple scales method (MSM), is introduced which homogenises the complex windings domain and allows for the estimation of its effective thermal properties. The homogenisation through the MSM is performed solving a single elementary cell. The MSM also allows for the reconstruction of the actual thermal field. Extensive numerical and experimental validation is provided, in particular for the case of electrical windings encapsulated with epoxy. The thermal homogenisation is then combined with an electromagnetic homogenisation technique to estimate winding losses including AC effects, such as proximity and skin effects. The coupled analysis is validated numerically on reference test problems, and experimentally, on a suitably built "motorette". The method is proven to correctly predict losses including thermal effects and to estimate magnitude and location of the temperature hotspot within the winding domain. This work also introduces a new approach for building thermal equivalent circuits that represents the most commonly employed modelling technique for electrical machine thermal analysis. Here the TEC approaches are thoroughly analysed, highlighting limitations. The proposed new technique extends the range of numerical accuracy, accounting for high Biot numbers (up to Bi = 2) and internal heat generation. The result of this approach is higher spatial resolution about the temperature field within the winding domain and thus enables improved information on hotspot location and magnitude. The method is experimentally validated and also applied to model an electrical machine for full-electric in-wheel vehicle propulsion.
2 citations
17 Jun 2001
TL;DR: This work presents a procedure valid to solve small signal tests of magnetic components (inductors and multiwinding transformers) by means of analytical expressions based on the use of the transmission line 1D model of the magnetic component.
Abstract: This work presents a procedure valid to solve small signal tests of magnetic components (inductors and multiwinding transformers) by means of analytical expressions. The method is based on the use of the transmission line 1D model of the magnetic component. Since the test can be analytically solved, the procedure allows skipping the use of electrical simulators in order to solve the small signal analysis. The procedure is valid for general cases, and therefore it can be implemented using mathematical packages (Mathcad, Madab, Mathematica, etc.) or using programming languages. The method has been validated by means of comparisons with actual measurements as well as FEA solvers results.
2 citations
TL;DR: In this paper, a multurn-coil system was designed to measure magnetic susceptibility up to 1 MHz with a field strength more than 10 Oerms, and a typical coil was inserted inside the induction coil to provide a spatially-controllable field distribution.
Abstract: Some practical applications of magnetic nanoparticles in theranostics systems require a preliminary study on magnetically-induced particle behavior with regard to the frequency and field strength-dependent magnetic susceptibility, measurable through a complex magnetic susceptometer. A wide measurement range, however, is favorable to provide comprehensive understanding in relaxation dynamics of magnetic nanoparticles, as well as their nonlinear magnetization. Therefore, we particularly designed a coil system, thus achieving measurability of complex magnetic susceptibility up to 1 MHz with field strength more than 10 Oerms. Our design includes parallel-arranged multiturn-coils as an induction coil to minimize overall resistance without a significant decrease of coil inductance, thus improving current-to-field conversion efficiency with a minimized dimension. Additionally, a typical coil was inserted inside the induction coil to provide a spatially-controllable field-distribution, and a single-turn sheet-coil was used as a sensor to obtain minimum noise and improve sensitivity reaching a relatively-constant value regardless field strength and temperature.
2 citations
TL;DR: In this paper, three potential options to implement multicore planar transformers using a printed circuit board as windings are presented and compared using finite-element analysis of the electromagnetic fields.
Abstract: Three potential options to implement multicore planar transformers using a printed circuit board as windings are presented The first option utilises a distributed transformer configuration where smaller transformer elements with separate cores are interconnected to function as a single transformer The second option has a lumped core configuration which uses multiple cores arranged to function as a monolithic core while the third is a hybrid of the first two configurations The three transformers were designed for a 400/40 V, 1 kW, DC-DC converter application at a switching frequency of 100 kHz The electrical and thermal characteristics of the three configurations are compared using finite-element analysis of the electromagnetic fields The simulation results are also validated by measuring the electrical characteristics of transformers fabricated in the three configurations The electrical and thermal characteristics of the transformers are also verified in a power converter system Results of the comparison show that the distributed core configuration gave the best thermal characteristic while the lumped core configuration has the least winding losses
2 citations
References
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TL;DR: In this article, a multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers.
Abstract: IN any multilayer winding carrying an alternating current, such as the windings illustrated in figures 1, 2, and 3, each layer of copper lies in the alternating magnetic field set up by the current in all the other layers. Eddy currents are set up in each layer in a direction to partly neutralize the magnetic intensities in the interior of the copper wire in each layer. As a result of the eddy-current losses in the copper, the effective resistance of the winding to the alternating current it carries may be many times its resistance to continuous currents.
103 citations
TL;DR: In this article, the authors discuss the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron, and propose a method to identify the most important causes.
Abstract: The object of the present paper is the discussion of the more important causes of eddy currents in heavy conductors carrying alternating currents and surrounded on three sides by iron.
93 citations
64 citations
TL;DR: In this article, it is shown that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance.
Abstract: It is well-known that a considerable proportion of the effective resistance of inductive coils when used at radio frequencies is caused by the eddy-currents set up in the wires of the coils by the alternating magnetic field in which they are situated, and that in extreme cases the alternating current resistance may amount to more than one hundred times the direct current resistance. It is therefore important to have reliable formulae for the eddy-current resistance of such coils in order to determine the conditions which will reduce the eddy-current losses to a minimum. The simplest case, that of a long straight cylindrical wire under the action of its own current, has been treated by Kelvin, Rayleigh, Heaviside, and others. The general effect is known as the “skin effect,” because the current tends to concentrate more and more upon the skin of the conductor as the frequency increases.
49 citations
TL;DR: In this article, the authors show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots.
Abstract: The principal object of this paper is to show how hyperbolic functions of complex angles may be applied to the solution of the problem of heat losses in rectangular conductors that are embedded in open slots. A certain knowledge of the functions themselves is presupposed. Inasmuch, however, as they are handled like trigometric functions of real angles?except in regard to the plus and minus signs?it is a simple matter to acquire the requisite technical skill to use them. The hyperbolic function of a complex angle, consisting as it does of a real and an imaginary part, may represent a vector?the real part being the component of the vector along the horizontal, and the imaginary part, component along the vertical. Thus, for example, A sinh (x + j x) represents a vector just as A e j ? A/?, A (cos ? + j sin ?) represent vectors. Considerable experience has shown that the vector method for handling a-c. problems is much superior to the original method in which simple trigonometric functions were used. With this lesson before us, it should require but little contact with the problem at hand to demonstrate the superiority of the vector method, even though it employs the possibly unfamiliar hyperbolic quantities. These hyperbolic vectors have been used for a number of years in the analysis of problems involving a-c. circuits, which have distributed inductance and capacitance, and have proved their usefulness.
27 citations